#6807933

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42648	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(6,9)(7,11)(13,14,19,24)(15,20,27,35)(16,21,30,38)(17,22,26,34)(18,23,29,37)(25,33,42,44)(28,36,31,39)(32,40,41,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,24,35)(16,28,30,31)(17,29,26,18)(21,36,38,39)(22,37,34,23)(25,41,42,32)(33,43,44,40);(2,4)(13,16,29,41,19,30,18,32)(14,21,37,43,24,38,23,40)(15,25,17,31,27,42,26,28)(20,33,22,39,35,44,34,36);(5,7,8,11)(6,10,9,12)(16,30)(21,38)(25,42)(28,31)(32,41)(33,44)(36,39)(40,43);(6,9)(10,12)(13,17,18,15,19,26,29,27)(14,22,23,20,24,34,37,35)(16,31,32,25,30,28,41,42)(21,39,40,33,38,36,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715129);